7. Special Cases
Society for Industrial and Applied Mathematics eBooks · 1999
In this chapter we discuss some important special cases. In particular, the cases considered have special features due to the problem data that simplify the construction of the controller. A class of bilinear systems is discussed in §7.1. These systems are linear systems with an additional term in the state dynamics consisting of a product of the state and control. The information state for this system is finite dimensional, given by a driven Riccati equation, and the dynamic programming PDE, which determines the controller, is correspondingly defined on a finite-dimensional space. Linear systems are special cases of this, and the controller is determined by a Riccati equation. Thus the well-known linear H∞ controller is obtained from the information state theory as a special case. In section 7.3 systems that satisfy the certainty equivalence principle are discussed. Here, the controller is constructed from a pair of PDEs defined on a finite-dimensional space. 7.1 Bilinear Systems Consider the following class of bilinear systems: Gbilinear : { x ˙ =A (u) x+ B1 w+ B2 u,z= C1 x+ D12 u,y= C2 x+ D21 w. 7.1 Here, we take the control input dimension for simplicity, and A(u)=A+ B3 u. In our general notation (Chapter 2), would read , in terms of the notation of (7.1). The presence of this term means that the dynamics have a multiplicative term involving both x and u—hence the term bilinear. The matrices are of the appropriate sizes (e.g., is n×n), and the standard assumptions used earlier apply, except that is no longer bounded.